Sounds like your dear hubby is having an emotional affair with this girl.. emotional affairs do take place with out sexual contact involved. It could be that your husband has issues he is unwilling to face or deal with at home, thus hidding behind an emotional affair with this other lady.
Trust is a huge issue in any marriage, to regain that broken trust, the other person must come clean in all aspects, stop communication with the other person in the affair, and focus back on the marriage 100%.